Exact solutions on the defored Hermitian-Yang-Mills (dHYM) equation with the quantum gravity

Using the Courant algebroid (CA) and geometric invariant theory (GIT), I give totally analytical exact solutions on the dHYM equation:

- sin(θ) (ω² - α²) + 2cos(θ) ωΛα = 0,

which can be written:

(cot(θ)ω + α)² = (1 + cot²(θ))ω².

S. Lukyanov and A. B. Zamolodchikov (1996) that the expectation value of the composite field TT , built from the components of the energy-momentum tensor, is expressed exactly through the expectation value of the energy-momentum tensor itself. They proposed a sine-Gordon model is defined by the Euclidean action (QFT):

Α_SG= ∫d²x {1/(16π) (∂_νφ)² - 2μ cos(βφ)}, where β, μ are parameters. The lightest  of bound states coincides with the particle associated with the field φ in perturbative treatment of the QFT. Its mass m is given by

m = 2M sin(πξ/2).

In this paper I will prove that, in a maximal complex torus, μ =1, M= 0.5, then there exisits a true topological quantum gravity system with an effective QFT:

m = sin(πξ/2) = √(1 - V/4), V =[ 2cos(πξ/2)] = 4(1 - m²)  and

the momentum mV= sin(πξ/2) × 4(cos²(πξ/2)) = 2cos(πξ/2) × sin(πξ) = 4 × m(1 - m²) then the GNL's kernel: GmV = 1, or 4 × Gm(1+m)(1-m) = 1. It is clear that:

m(1+m)(1-m) =1/(4G). As a special example, m = 1/2. V= (√3)² = 3 ... km/s (light velocity or photon velocity in quasi- stable state), G = G₀ = 2/3 = 0.666666..., i.e. the classical Newton's gravity constant.

The final fact we wish to record is less-known:

2n ± √3 + 1/(2n ± √3) |_n=1 = 4;

2n ± √5 -  1/(2n ± √5) |_n=1 = 4.

This important result on the extended Courant algebroid is just ergodic theory of the dHYM solutions, and a crucial point in the GNL.

This Extended Courant algebroid (CA) proves an important physical fact: the maximal perturbative velocity of photon or extreme velocity of particle equal to 4, at this quantum state, the mass of particle m_limit = √(1 - 4/4) = 0. This study result on quantum gravity by the dHYM model will induce a new theoretical physics, which includes the SR or GR and modern quantum mechanics.